Manhattan distance

The Manhattan distance is a different way of measuring distance. It is named after the grid shape of streets in Manhattan. If there are two points, <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math>, the Manhattan distance between the two points is <math>|x_1 - x_2| + |y_1 - y_2|</math>.

This distance can be imagined as the length needed to move between two points in a grid where you can only move up, down, left or right.

Extension

This definition can be used for three and higher dimensions too. If there are two vectors, <math>\mathbf{p}=(p_1,p_2 \dots, p_n)</math> and <math>\mathbf{q}=(q_1,q_2 \dots, q_n)</math>, then the manhattan distance between the two points is the absolute value of the difference between all numbers in the vector. Or, in notation: <math display="block">\sum^n_{i=1}|p_i-q_i| </math>